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Wikipedia / Linear prediction

Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples.

In digital signal processing, linear prediction is often called linear predictive coding (LPC) and can thus be viewed as a subset of filter theory. In system analysis, a subfield of mathematics, linear prediction can be viewed as a part of mathematical modelling or optimization.

The prediction model

The most common representation is

the predictor coefficients. The error generated by this estimate is

is the true signal value.

are chosen.

For multi-dimensional signals the error metric is often defined as

is a suitable chosen vector norm are routinely used within Kalman filters and smoothers to estimate current and past signal values, respectively, from noisy measurements.

Estimating the parameters

is the root mean square criterion which is also called the autocorrelation, which yields the equation

for 1 ≤ j p, where R is the autocorrelation of signal xn, defined as

  1. ,

and E is the expected value. In the multi-dimensional case this corresponds to minimizing the L2 norm.

The above equations are called the normal equations or Yule-Walker equations. In matrix form the equations can be equivalently written as

Toeplitz matrix, the parameter vector.

Another, more general, approach is to minimize the sum of squares of the errors defined in the form

.

On the other hand, if the mean square prediction error is constrained to be unity and the prediction error equation is included on top of the normal equations, the augmented set of equations is obtained as

matrix.

Specification of the parameters of the linear predictor is a wide topic and a large number of other approaches have been proposed. In fact, the autocorrelation method is the most common and it is used, for example, for speech coding in the GSM standard.

is computationally a relatively expensive process. The Gaussian elimination. A faster algorithm is the Levinson recursion proposed by Norman Levinson in 1947, which recursively calculates the solution, running in time (vs for Gaussian elimination). In particular, the autocorrelation equations above may be more efficiently solved by the Durbin algorithm.

In 1986, Philippe Delsarte and Y.V. Genin proposed an improvement to this algorithm called the split Levinson recursion, which requires about half the number of multiplications and divisions. terms.

Another way of identifying model parameters is to iteratively calculate state estimates using Kalman filters and obtaining maximum likelihood estimates within expectation–maximization algorithms.

For equally-spaced values, a polynomial interpolation is a linear combination of the known values. are given by the corresponding row of the triangle of binomial transform coefficients. are

See also

References

Further reading